Appendix VI. 1087 Change of Pole. In Fig. 961 are shewn six forces exactly balancing, as proven from the vector figure. Let the link polygon be drawn in the first place for a pole 0, and let it be desired to re-draw it for a new pole Or This has been done point by point in the usual manner, arriving at the dotted polygon, but could have been effected more simply by the use of a • simple property connected with two poles of the same vector figure. Join O Ox producing both ways. Next take any line of one link polygon and the corresponding line on the other link polygon; produce these till they meet, obtaining, say, the point <?3. Doing this for all the corresponding sets of lines of the polygons, we have the additional points ^ <?0 ^. It will now be found that these points are in one straight line, that is itself parallel to the lines O Or For proof: ^ ^ 01 = oO-fOi also, 01 = oC^-hC^i But because 01 balances 10, oO-r-Oi balances C^o-}- iOl and oO-f-Oi+OjO-J-iC^ = zero But OjO+oO = O O1 and Oi-f iOx =0^0 The force O Oj is therefore the resultant of the corresponding link pairs in either link polygon. Its reciprocal el <?0 must join the meets of the link pairs, and must be parallel to O Or A similar treatment of other pairs will prove the line <?3 to e± also parallel to 0 Ox. Use of Link Polygon.—As an example take a semicircular roof, of which half is shewn at A B c, Fig. 962. Let the wind blow from right to left, and we require to know the direction and magnitude of the resultant of all the pressure that comes upon c B, in order to carry out the construction given at p. 470, and thus arrive at the stresses on the members. Divide the arc B c into any convenient number of equal parts at r>, E, F, G, H, j, K, L, in this case nine. Assume the pressure intensity uniform over each division, though varying from one division to anpther, and let each piece of area acted upon be H x br, where K is the height and b' the breadth between the bays, both in feet. If the wind intensity be w, the normal intensity n will vary from w at B to zero at c, and its value at any point can be found from the